Question

A dance class consists of $22$students, of which $10$are women and 12 are men. If $5$men and $5$women are to be

chosen and then paired off, how many results are possible?

Short answer

The possible results are$23,950,080$.

Step-by-step solution

Step 1. Given information.

Total number of students $=22$

No. of men $=12$

No. of women$=10$

Step 2. Find the number of possible results.

No. of ways of selecting $5$men out of $12$= $\frac{12!}{5!7!}=\frac{12\times 11\times 10\times 9\times 8\times 7!}{5\times 4\times 3\times 2\times 1\times 7!}=792$

No. of ways of selecting $5$men out of $10$= $\frac{10!}{5!5!}=\frac{10\times 9\times 8\times 7\times 6\times 5!}{5\times 4\times 3\times 2\times 1\times 5!}=252$

No. of ways that $5$men and $5$women can be paired off = $5!=5\times 4\times 3\times 2\times 1=120$

Therefore, the total number of possible results are =$792\times 252\times 120=23,950,080$